Domino tilings of threedimensional regions:
flips and twists
Abstract
In this paper, we consider domino tilings of regions of the form , where is a simply connected planar region and . It turns out that, in nontrivial examples, the set of such tilings is not connected by flips, i.e., the local move performed by removing two adjacent dominoes and placing them back in another position. We define an algebraic invariant, the twist, which partially characterizes the connected components by flips of the space of tilings of such a region. Another local move, the trit, consists of removing three adjacent dominoes, no two of them parallel, and placing them back in the only other possible position: performing a trit alters the twist by . We give a simple combinatorial formula for the twist, as well as an interpretation via knot theory. We prove several results about the twist, such as the fact that it is an integer and that it has additive properties for suitable decompositions of a region.
1 Introduction
Tiling problems have received a lot of attention in the second half of the twentieth century, twodimensional domino and lozenge tilings in particular. For instance, Kasteleyn [Kasteleyn19611209], Conway [conway1990tiling], Thurston [thurston1990], Elkies, Propp et al. [jockusch1998random, cohn1996local, elkies1992alternating], Kenyon and Okounkov [kenyonokounkov2006dimers, kenyonokounkov2006planar] have come up with very interesting techniques, ranging from abstract algebra to probability. More relevant to the discussion in this paper are the problems of flip accessibility (e.g., [saldanhatomei1995spaces]).
Attempts to generalize some of these techniques to the threedimensional case were made. The problem of counting domino tilings, even of contractible regions, is known to be computationally hard (see [pak2013complexity]), but some asymptotic results, even for higher dimensions, date as far back as 1966 (see [hammersley1966limit, ciucu1998improved, friedland2005theory]). In a different direction, some “typically twodimensional” properties were carried over to specific families of threedimensional regions (see [randall2000random, linde2001rhombus, bodini2007tiling]).
Others have considered difficulties with connectivity by local moves in dimension higher than two (see, e.g., [randall2000random]). We propose an algebraic invariant that could help understand the structure of connected component by flips in dimension three.
In this paper, we investigate tilings of contractible regions by domino brick pieces, or dominoes, which are simply rectangular cuboids. An example of such a tiling is shown in Figure 1. While this 3D representation of tilings may be attractive, it is also somewhat difficult to work with. Hence, we prefer to work with a 2D representation of tilings, which is shown in Figure 4.
A key element in our study is the concept of a flip, which is a straightforward generalization of the twodimensional one. We perform a flip on a tiling by removing two (adjacent and parallel) domino bricks and placing them back in the only possible different position. The removed pieces form a slab, in one of three possible directions (see Figure 4). The flip connected component of a tiling of a threedimensional region is the set of all tilings of that can be reached from after a sequence of flips. It turns out that for large regions the number of flip connected components is also large, but some of them may contain many tilings. One of the aims of this paper is to study such connected components.
As in [primeiroartigo], we also consider the trit, which is a move that happens within a cube with two opposite “holes”, and which has an orientation (positive or negative). More precisely, we remove three dominoes, no two of them parallel, and place them back in the only other possible configuration (see Figure 4).
A cylinder is a region of the form (possibly rotated), where is a simply connected planar region with connected interior. In this paper, we introduce an algebraic invariant, the twist , defined in Section 3 for tilings of a cylinder.
In [primeiroartigo], we study cylinders with , called duplex regions. Although they are related to the general theory, tilings of these regions have some interesting characteristics of their own; in particular, we can define a polynomial for tilings of duplex regions which is invariant by flips and which is finer than the twist. However, this construction breaks down when the duplex region is embedded in a region with more floors (see [primeiroartigo] for details).
Theorem 1.
Let be a cylinder, and a tiling of . The twist is an integer with the following properties:

[label=()]

If a tiling is reached from after a flip, then .

If a tiling is reached from after a single positive trit, then .

If is a duplex region, then for any tiling of .

Suppose a cylinder , where each is a cylinder (they need not have the same axis) and such that . Then there exists a constant such that, for any family , a tiling of ,
The definitions of twist are somewhat technical and involve a relatively lengthy discussion. We shall give two different but equivalent definitions: the first one, given in Section 3, is a sum over pairs of dominoes. At first sight, this formula gives a number in and depends on a choice of axis. However, it turns out that, for cylinders, this number is an integer, and different choices of axis yield the same result. The proof of this claim will be completed in Section 6, and it relies on the second definition, which uses the concepts of writhe and linking number from knot theory (see, e.g., [knotbook]).
One can ask whether the twist can be extended to a broader class of regions. The following (see [tese]) holds: let be a simply connected region (not necessarily a cylinder), and be two tilings of . Suppose is a box and is a tiling of (it is not true for arbitrary regions that and exist). Define : this turns out to depend neither on the choice of box nor on the choice of tiling . Therefore, if we choose a base tiling and define , then satisfies items 1 and 2 in Theorem 1. Different choices of base tiling only alter the twist by an additive constant.
In addition to the combinatorial and knottheoretic interpretations developed in this article, it is also possible to give homological interpretations for the twist. These homological constructions are reminiscent of the twodimensional height functions (see [thurston1990]), although they behave more like “height forms”. The concept of flux (or flow), as in [saldanhatomei1995spaces] and [saldanhatomeiAnnuli], also becomes relevant. Although we will not discuss these constructions here, this homological point of view inspired many of our definitions.
One might also ask what the possible values for the twist of a certain region are. Some results in this direction are proved in [tese]: for instance, it turns out that, for a box of dimensions with (and even), the maximum possible value for the twist is of the order of .
The present paper is structured in the following manner: Section 2 introduces some basic definitions and notations that will be used throughout the paper. In Section 3, we define the invariant for cylinders, and prove its most basic properties. In Sections 4, 5, 6 and 7, we present different aspects of a connection between the twist of tilings and a few classical concepts from knot theory. Section 4 contains the “topological groundwork”, which consists of a number of definitions and results that help establish topological interpretations of the twist, and which are extensively used in the sections that follow it. In Section 5, we introduce a different formula for the twist of cylinders, and show that this new formula allows us to prove (once again, via topology) that the twist must always be an integer. In Section 6, we prove that the value of the twist of cylinders does not depend on the choice of axis, which is one of the main results in the paper. In Section 7, we discuss additive properties of the twist, and prove item 4 in Theorem 1. Finally, Section 8 contains some examples and counterexamples that help illustrate the theory.
This paper closely corresponds to part of the first author’s PhD thesis [tese]; the authors thank the examination board for helpful comments and suggestions. The authors are also thankful for the generous support of CNPq, CAPES and FAPERJ (Brazil).
2 Definitions and Notation
This section contains general notations and conventions that are used throughout the article, although definitions that involve a lengthy discussion or are intrinsic of a given section might be postponed to another section.
If is an integer, will denote (in music theory, is a half tone higher than in pitch). We also define to be the set .
Given , denotes the determinant of the matrix whose th line is , . If is a basis, write .
We denote the three canonical basis vectors as , and We denote by the set of canonical basis vectors, and . Let be the set of positively oriented bases with vectors in .
A basic cube is a closed unit cube in whose vertices lie in . For , the notation denotes the basic cube , i.e., the closed unit cube whose center is ; it is white (resp. black) if is even (resp. odd). If , define , or, in other words, if is black and if is white. A region is a finite union of basic cubes. A domino brick or domino is the union of two basic cubes that share a face. A tiling of a region is a covering of this region by dominoes with pairwise disjoint interiors.
We sometimes need to refer to planar objects. Let denote either or a basic plane contained in , i.e., a plane with equation , or for some . A basic square in is a unit square with vertices in (if ) or . A planar region is a finite union of basic squares.
A region is a cubiculated cylinder or multiplex region if there exist a basic plane with normal vector , a simply connected planar region with connected interior and a positive integer such that
we usually call a cylinder for brevity. The cylinder above has base , axis and depth . For instance, a cylinder with axis and depth can be written as , where . A cylinder means a cylinder with axis . A duplex region (see [primeiroartigo]) is a cylinder with depth .
We sometimes want to point out that the hypothesis of simple connectivity (of a cylinder) is not being used: therefore, a pseudocylinder with base , axis and depth has the same definition as above, except that the planar region is only assumed to have connected interior (and is not necessarily simply connected).
A box is a region of the form , where . Boxes are special cylinders, in the sense that we can take any vector as the axis. In fact, boxes are the only regions that satisfy the definition of cylinder for more than one axis.
Regarding notation, Figures 4, 4 and 4 were drawn with in mind. However, any allows for such representations, as follows: we draw as perpendicular to the paper (pointing towards the paper). If , we then draw each floor as if it were a plane region. Floors are drawn from left to right, in increasing order of .
The flip connected component of a tiling of a region is the set of all tilings of that can be reached from after a sequence of flips.
Suppose is a tiling of a region , and let , with . Suppose contains exactly three dominoes of , no two of them parallel: notice that this intersection can contain six, seven or eight basic cubes of . Also, a rotation (it can even be a rotation, say, in the plane), can take us either to the left drawing or to the right drawing in Figure 5.
If we remove the three dominoes of contained in , there is only one other possible way we can place them back. This defines a move that takes to a different tiling by only changing dominoes in : this move is called a trit. If the dominoes of contained in are a plane rotation of the left drawing in Figure 5, then the trit is positive; otherwise, it’s negative. Notice that the sign of the trit is unaffected by translations (colors of cubes don’t matter) and rotations in (provided that these transformations take to ). A reflection, on the other hand, switches the sign (the drawing on the right can be obtained from the one on the left by a suitable reflection).
3 The twist for cylinders
For a domino , define to be the center of the black cube contained in minus the center of the white one. We sometimes draw as an arrow pointing from the center of the white cube to the center of the black one.
For a set and , we define the (open) shade of as
where denotes the interior of . The closed shade is the closure of . We shall only refer to shades of unions of basic cubes or basic squares, such as dominoes.
Given two dominoes and of , we define the effect of on along , as:
In other words, is zero unless the following three things happen: intersects the shade of ; neither nor are parallel to ; and is not parallel to . When is not zero, it’s either or depending on the orientations of and .
For example, in Figure 6, for , the yellow domino has no effect on any other domino: for every domino in the tiling. The green domino , however, affects the two dominoes in the rightmost floor which intersect its shade, and for both these dominoes.
If is a tiling, we define the pretwist as
For example, the tiling on the left of Figure 4 has pretwist equal to . To see this, notice that each of the four dominoes of the leftmost floor that are not parallel to has nonzero effect along on exactly one domino of the rightmost floor, and this effect is in each case. The reader may also check that the pretwist of the tiling in Figure 6 is .
Lemma 3.1.
For any pair of dominoes and and any , . In particular, for a tiling of a region we have .
Proof.
Just notice that if and only if and . ∎
Translating both dominoes by a vector with integer coordinates clearly does not affect , as . Therefore, if is a tiling and , where , then .
Lemma 3.2.
Let be a region, and let . Consider the reflection ; notice that is a region. If is a tiling of and , then the tiling of satisfies .
Proof.
Given a domino of , notice that and that . Therefore, and
Therefore, and thus . Since , Lemma 3.1 implies that , completing the proof. ∎
A natural question at this point concerns how the choice of affects . It turns out that it will take us some preparation before we can tackle this question.
Proposition 3.3.
If is a cylinder and is a tiling of ,
This result doesn’t hold in pseudocylinders or in more general simply connected regions; see Section 8 for counterexamples.
Definition 3.4.
For a tiling of a cylinder , we define the twist as
Let , and let be such that . A region is said to be fully balanced with respect to if for each square , where and , each of the two sets and contains as many black cubes as white ones. In other words,
is fully balanced if it is fully balanced with respect to each .
Lemma 3.5.
Every pseudocylinder (in particular, every cylinder) is fully balanced.
Proof.
Let be a pseudocylinder with base and depth , let and let , where is such that and . Consider .
If is the axis of the pseudocylinder, then , for some square and some . Now , which clearly contains black cubes and white ones; similarly, contains black cubes and white ones.
If is perpendicular to the axis of the pseudocylinder, assume without loss of generality that is the axis. Let denote the orthogonal projection on , and let , which are planar regions, since they are unions of squares of . If , we have , which clearly has the same number of black squares as white ones. ∎
Proposition 3.6.
Let be a region that is fully balanced with respect to .

[label=(),topsep = 0.1px]

If a tiling of is reached from after a flip, then

If a tiling of is reached from after a single positive trit, then .
Proof.
In this proof, points towards the paper in all the drawings. We begin by proving 1. Suppose a flip takes the dominoes and in to and in . Notice that and . For each domino , define
Notice that
Case 1.
Either or is parallel to .
Assume, without loss of generality, that (and thus also ) is parallel to . By definition, for each domino . Now notice that and are parallel and in adjacent floors (see Figure 7) : since , it follows that for each domino , so that and thus .
Case 2.
Neither nor is parallel to .
In this case, for some square of side and normal vector .
Notice that ; let .
Let be a domino that is completely contained in : we claim that . This is obvious if is parallel to ; if not, we can switch the roles of and if necessary and assume that is parallel to , which implies that . Now notice that is in the shades of both and , so that . Hence, if (or if ), .
For dominoes that intersect but are not contained in it, first observe that by switching the roles of and and switching the colors of the cubes (i.e., translating) if necessary, we may assume that the vectors are as shown in Figure (a)a. By looking at Figure (b)b and working out the possible cases, we see that
Now for such dominoes, points away from the region if and only if intersects a white cube of , and points into the region if and only if intersects a black cube in : hence,
because is fully balanced with respect to . A completely symmetrical argument shows that , so we are done.
We now prove 2. Suppose is reached from after a single positive trit. By rotating and in the plane (notice that this does not change ), we may assume without loss of generality that the dominoes involved in the positive trit are as shown in Figure 5. Moreover, by translating if necessary, we may assume that the vectors are as shown in Figure 9.
A trit involves three dominoes, no two of them parallel. Since dominoes parallel to have no effect along , we consider only the four dominoes involved in the trit that are not parallel to : , and . Define with the same formulas as before.
By looking at Figure 5, the reader will see that and .
Let : is shown in Figure 9. contains a single square of side and normal vector . Define , and notice that (see Figure 9) , where are three basic cubes: if we look at the arrows in Figure 9, we see that two of them are white and one is black. Since is fully balanced with respect to ,
By looking at Figure 9, we see that we have a situation that is very similar to Figure (b)b; for each , we have
(when we say that points into or away from , we are assuming that intersects one cube of ). Hence,
A completely symmetrical argument shows that , and hence
which completes the proof. ∎
4 Topological groundwork for the twist
In this section, we develop a topological interpretation of tilings and twists. Dominoes are (temporarily) replaced by dimers, which, although formally different objects, are really just a different way of looking at dominoes. Although we will tend to work with dimers in this and the following section, we may in later sections switch back and forth between these two viewpoints.
Let be a region. A segment of is a straight line of unit length connecting the centers of two cubes of ; in other words, with , where and are the centers of two cubes that share a face: this segment is a dimer if is the center of a white cube. We define (compare this with the definition of for a domino ). If is a segment, denotes the segment : notice that either or is a dimer.
Two segments and are adjacent if (here we make the usual abuse of notation of identifying a curve with its image in ); nonadjacent segments are disjoint. In particular, a segment is always adjacent to itself.
A tiling of by dimers is a set of pairwise disjoint dimers such that the center of each cube of belongs to exactly one dimer of . If is a tiling, denotes the set of segments .
Given a map , a segment and an integer , we abuse notation by making the identification if for each . A curve of is a map such that is (identified with) a segment of for . We make yet another abuse of notation by also thinking of as a sequence or set of segments of , and we shall write to denote that for some .
A curve of is closed if ; it is simple if is injective in . A closed curve of is called trivial: notice that, in this case, (when identified with their respective segments of ). A discrete rotation on is a function with , for a fixed . If and are two closed curves, we say if and for some discrete rotation on .
Given two tilings and , there exists a unique (up to discrete rotations) finite set of disjoint closed curves such that and such that every nontrivial is simple. Figure 12 shows an example. We define .
Translating effects from the world of dominoes to the world of dimers is relatively straightforward. For , will denote the orthogonal projection on the plane . Given two segments and , we set:
Notice that this definition is analogous to the one given in Section 3 for dominoes.
The definition of is given in terms of the orthogonal projection . From a topological viewpoint, however, this projection is not ideal, because it gives rise to nontransversal intersections between projections of segments. In order to solve this problem, we consider small perturbations of these projections.
Recall that is the set of positively oriented basis with vectors in . If and , will be used to denote the projection on the plane whose kernel is the subspace (line) generated by the vector . For instance, if is the canonical basis, . Notice that is the orthogonal projection on the plane , and, for small , is a nonorthogonal projection on which is a slight perturbation of .
Given , and small nonzero , set the slanted effect
Recall from knot theory the concept of crossing (see, e.g., [knotbook, p.18]). Namely, if ,